A-infinity algebras, strand algebras, and contact categories

(83 pages) – on the arXiv.

Abstract: In previous work we showed that the contact category algebra of a quadrangulated surface is isomorphic to the homology of a strand algebra from bordered Floer theory. Being isomorphic to the homology of a differential graded algebra, this contact category algebra has an A-infinity structure. In this paper we investigate such A-infinity structures in detail. We give explicit constructions of such A-infinity structures, and establish some of their properties, including conditions for the nonvanishing of A-infinity operations. Along the way we develop several related notions, including a detailed consideration of tensor products of strand diagrams.


Is the traditional mathematics blackboard lecture dead?

The Australian Mathematical Society Annual Meeting this year included a public debate on the topic “Is the traditional mathematics blackboard lecture dead?” I was on the affirmative team, arguing that the traditional blackboard lecture is in fact dead. Below is some approximation to my remarks. Being a case for one side of the argument, and in a context of an event as much for entertainment as for serious discussion, the below is a small part of my views on the matter.

We lost the debate rather convincingly — and arguing against blackboard lectures to mathematicians is a rather unpopular cause! — but nonetheless it was an entertaining event raising some important issues for tertiary mathematics educators. I thank my colleagues Birgit Loch and Marty Ross for their valiant prosecution of this unpopular cause; congratulations to Adrianne Jenner, Heather Lonsdale and John Roberts on their victory; and to Adam Spencer for moderating.

* * *

We’re here to debate whether the traditional blackboard mathematics lecture is dead.

We are asking whether something despised, deserted, and largely replaced, is dead. Is a corpse dead? Yes, a corpse is dead. Nonetheless, we can still debate the question.

Now I would dearly love this corpse to be resurrected — at least, in its better forms. But resurrection is well beyond my pay grade, and I look forward to my colleagues on the other side bringing forth its second coming.

In our affirmative case, I‘ll be laying out the issues: what the traditional blackboard lecture is, and how it’s dead or dying.

Birgit will provide the proof around attendance, technology, and how traditional blackboard lectures have been replaced.

And Marty will be summing up and sledging the opposition.

* * *

So what does a traditional mathematical blackboard lecture look like?

The students shuffle in, notebooks open. Some of them have understood the material. Most have not — we all know this, because we mark their exams. The lecturer mumbles incomprehensibly into the blackboard. The students are bored; probably the lecturer too. The lecturer copies their notes onto the blackboard. The students copy the notes from the blackboard into their notebooks.

In this way, traditional mathematics blackboard lectures are a transmission channel for maths notes. This gives maths notes a very good reproductive advantage. The same cannot be said for the lecturer.

* * *

Now, not all traditional backboard lectures are that bad, although as we all know, often they really are. A knowledgeable lecturer joyfully expounding the subject they know and love, in all its depth and beauty, can really shine.

But this is not what most people experience.

And how could it be – when most peoples’ experience is of first year units, where students enrol with imagination beaten out of them by a thousand algebra exercises, ever-decreasing background knowledge, ever-increasing financial stresses, ever-decreasing attention spans, and we have to complete their secondary education, fix their miseducation, and teach the actual intended content, if they ever turn up to lectures, in a tightly constrained timeframe, with limited resources, cramming in those fundamentals we’d be embarrassed for students not to know in the no time left before the end of semester?

The tragedy is that for most students, this is our only chance to get them into mathematics, and we lose them. They don’t learn much – even what mathematics is. And those are the ones who pass.

These circumstances make a mockery of our goals as mathematics educators. How can we nurture that free creativity, that tightly constrained logic, that we recognise as the glory of mathematical thought – that joy at playing with new ideas and problems? No, in these circumstances, the traditional blackboard lecture reinforces all the worst of secondary schooling: regimented curriculum; passivity; boredom. But now at scale.

Tertiary education is a mass social institution. University maths departments teach thousands of students each semester. The bulk of these students have only ever taken lower level subjects where they’ve gotten a taste of the traditional blackboard lecture and then run a thousand miles away.

And it is this, the mass social phenomenon that we mean by the traditional blackboard lecture. It is this tradition which is dying. It is already dead. And there is no reason to mourn its demise.

* * *

Let me turn to traditional blackboard lectures as they currently exist, in my own experience, at Monash.

Well, they don’t — at the first year level. They are literally dead, as a matter of cruel hard fact. Large first year classes are taught in lecture theatres with no blackboards at all… or whiteboards, for that matter.

However… I don’t know if I should say this, but some blackboards actually remain at Monash.
There is a small holdout, a rebel building, camouflaged in 1970s mission brown. It is, so I’m told, slated for demolition. No doubt soon it will come into view of the death star. The building next door is already gone.

But in the meantime, traditional blackboard lectures are still taught on this holdout rebel planet. In these theatres — these arenas! — the chalk still flies, the dusters still dust, the blackboards sail up and down in banks of threes.

I lectured there as nearby buildings crumbled around me — literally. Lemmas were interrupted by jackhammers, propositions by demolitions, and proofs were built up as roofs came crashing down. Now there’s a postmodern metallic learning space next door.

All that remains, for us, of the traditional blackboard lecture, is this forlorn mission brown outpost. And it is really a metaphor for tertiary mathematics education today.

Support for traditional lectures has crumbled. Their time is past. And whatever we think of them, we need to think about what will take their place. Because if we don’t decide for ourselves, there are plenty of others willing to decide for us.

* * *

Now, just because something is dead does not mean it is bad. Evariste Galois is dead, but he was awesome. And so it is for traditional blackboard lectures: some were good; but most, in practice, were bad; and all are dead or dying.

What is to be done, and the Paradox of Choice

“What should I do?” This question is commonplace around, at least, the richer and more privileged parts of the world: there is so much wrong with the world, but I don’t know what to do. How to respond?

To a first approximation, this seems like a bad question. There is too much to do, not too little. The immediate answer seems to be: if you are asking this question, you are probably thinking too much. Stop asking, and get doing! There’s plenty to do, so do something.

But, to a second approximation, I think this question is quite an important one, and reveals some desperate failures of political philosophy and thinking across the world. It’s always worth taking some time to think about the broader picture, and how our actions fit into it.

* * *

My understanding is that this is a much more common question in the global North, in the richer western nations, than elsewhere. (I use the phrase “richer world” as a shorthand for people in the global north, the richer nations, the “developed” nations, though this is only a rough approximation and applies to anyone with a sufficiently privileged socio-economic position to have the freedom of some choice over what they do.) Chomsky often notes that, unlike elsewhere, he is never asked this question in the poorer parts of the planet, in the “third world”, the global south. And indeed, if oppression is clear and present, if you can, you go and do what needs to be done.

Broadly speaking, people in the richer world have more time, more resources, more wealth available to them. There is, on the whole, much more free time and excess cognitive capacity to figure out what to do. But there is no shortage of things gone wrong, of iniquities inflicted on other parts of the world. From climate change, to economic inequality, to civil liberties, to Orwellian surveillance, to creeping militarisation, to indigenous rights, to corporate power, to the erosion of democracy, to unjust wars and terrorism waged around the world — the litany is familiar.

Indeed, there is arguably *more* to fight against. To ask “What should I do?”, when there is so much to be done, rings shallow.

But there is also greater *distance* to the problems. The problems are further away, if only in a psychological sense, and lives are lived in a more comfortable consumer cocoon. The circumstances are no doubt familiar to most people reading this.

Wars are waged far away. Asylum seekers and refugees are locked up far away. Economic institutions by design, and by law, keep information about production, distribution and income far from public view. News media presents a skewed view of the world in keeping with their owners and largely consists of irrelevant distractions. Governments govern from far away, keeping their distance from the people. Never is a citizen given any impression that they can have any effect on any of it — except perhaps between a particular marketing effort called an “election” every few years. These problems, at least, then seem far away and it seems hard to have any effect on them.

Meanwhile, media, advertising, and custom combine to create a consuming culture of fear, guilt, infantilisation, anxiety, economic burden, and social apathy. Simultaneously marketing promotes instant gratification — and shame as to one’s own weight and self-image; unhealthy habits — and then guilt about it and products to relieve it; an array of consumer choices — but all equivalent brands of the same thing; beautiful houses to live in — and then non-stop wage slavery to pay off the debt; alienation from meaningful work — and then stress relief through shopping; “empowerment” by studying and taking approved positions of authority — while intimidating, overpowering and ridiculing those who take collective political action; a thin veneer of consumer goods — as a means to a thin veneer of happiness; turning fiction into “reality” television shows — and then turning reality into fiction at the news hour. There is fear of bad health, fear of not bringing up chidren properly, fear of boredom, fear of not having the latest consumer item — and then fear of terrorism, fear of foreigners, fear of Islam, fear of serial killers, fear of the latest enemy. It goes on. This dazzling, confusing, pervasive, terrifying, disorienting politico-economic-cultural system normalises apathy, renders political action barely imaginable, and alternative analyses of society barely thinkable. It is the bubble in which the richer world lives.

It may explain the lack of activity, but it is no excuse.

For those who have got to the point of asking the question, to some extent they have successfully resisted the consumer bubble. They are prepared to do something. They have found — provided they can navigate it critically — there is a huge amount of information about all the problems of the world, whether through learning, books, or (more usually) online. They have the sense that everything is wrong, and at least to some extent, know some details, though they are possibly overwhelming. But the barriers are still high, and the tendency to apathy all the more so.

This situation, very broadly speaking, entails that (a) citizens have relatively good access to information about what is going on, so that (b) they may more easily be are aware of the number and scale of things gone wrong and in need of fixing, while (c) living all too easily in a bubble of consumerism, marketing, media and superficiality, and (d) restrained by various social forces, from capitalist employment and debt to family ties and culture, so that knowledge of the scale and depth of the problem may combine with apathy and relative comfort and other social constraints to result in an outcome of little or no action. If the social outcome is no action, then the accumulation of knowledge and understanding, the asking of the question, was for nothing — it would, in fact, have been better to know nothing at all, since knowledge combined with inaction creates only stress and guilt, which are of negative value.

We are considering here the citizen’s point of view — the restraints and inhibitions upon them, to whatever extent voluntary, social or involuntary. In this view, the means and opportunity for doing something useful is just as important as the motive — though no doubt they all run together to some extent. This analysis is prior to any important particular issues such as climate change, civil liberties, workers’ rights, economic democracy, and so on — or any choice between them. (Of course there are many other valid analyses of political action, and how issues relate. Most notable in recent times is Naomi Klein’s argument that confronting climate change “Changes Everything”, so that in a certain sense it includes many other important social questions.) This is just another way of looking at it.

In any case, this means that to the extent that people are restrained from doing something useful by their job, their debts, and their culture, (i.e. as in (c) and (d) above), one objective of activism should be to minimise these restraints, by whatever struggle or action is appropriate. This includes maximising workplace rights, with the associated economic freedom it brings, and minimising debts, acting against exploitative financial institutions and mechanisms, and against the regressive influence of (at least) conservative religion. This much is straightforward.

And I don’t think many would argue that having more information, or being aware of what’s going on in the world, (i.e. (a) and (b) above) is a bad thing. In fact, on the contrary — if anything, people need more and better information. There is much to be said of how terrible the corporate media remains in reporting on important issues. There is much to be said of what is said, and what remains unsaid, in the corporate media. There is much to be said about the problems and issues facing “alternative” and online media. But I don’t think anyone could argue that the volume, or scale, of information is something that needs to be reduced. Perhaps perhaps less disinformation, better focused information, perhaps better presented, perhaps made less disorienting and more coherent, but not less actual information.

But the problem goes deeper. Even assuming that (a) citizens get good information about what’s going on, and (b) are highly socially aware, and (c) willing to break out of their superficial, consumer, media bubble, even against (d) any social restraints on their action, it does not seem to me to follow that many people would become effective change agents.

Why? There is still so much uncertainty about what to do. The problem of “too much information” becomes a “paradox of choice” paralysis.

So, to a first approximation, yes, the answer to “What should I do” is “stop thinking too hard and do something useful!” But to a second approximation, it probably is useful to have a way to make some sense of all the information thrown at us, the hundred slings and arrows of outrageous misfortune facing the world, the hundred different dimensions of outrageous misfortune and crime and tragedy and dysfunction.

To put it mathematically, the problem thrown at us has too many dimensions — it has the *curse of dimensionality*. We need to cut down the dimensions with a better analysis, order the data somehow, introduce an objective function or equivalent.

Yes, we need an objective.

* * *

The “paradox of choice” is a psychological phenomenon. When confronted with too many choices, the brain is overloaded. There is too much data to consider. Capacity is overwhelmed. Cognition is choked. There is too much to think through, too much uncertainty, and the response is paralysis. Nothing is done.

And indeed, one immediately observes this phenomenon in the situation facing the concerned citizen today: should I go to a climate rally? Volunteer for an environmental organisation? Participate in direct actions? Study to become a facilitator, human rights lawyer, scientist, renewable energy expert? Become a journalist, academic, social entrepreneur, hacker, engineer? Put my time into opposing the surveillance state, or international trade policy, or Aboriginal deaths in custody, or a hundred other things?

And, if we are to be honest, in many cases these are imponderable questions, on a par with “what shall I do with my life?” Except in rare cases, they are answered by some combination of one’s talents, pre-existing interests, people one meets, ideas one is exposed to, accidents of history, by fumbling through, choosing what seems best at the time, each time.

Even if we restrict to the question of what to do today, or this week, or in the next few months, there are too many problems, there is too much to do, there are too many problems, too much to learn about each of them, too much thinking to decide what to do. The result is too much time spent thinking and doing nothing, too much uncertainty about the best thing to do. It is exhausting to learn about one problem, let alone many. After learning, perhaps having a conversation or two, an argument or two, perhaps even a burst of social media, exhaustion sets in. The outcome again being nothing except more stress and guilt.

The mantra of direct action in this context, which cuts through the bullshit and gets something done, seems excellently wise.

* * *

The paradox of choice, while representing part of the problem facing the informed citizen, does not quite seem to entirely capture the issues. It is a good description of the consumer facing thirty brands of breakfast cereal — but there are extra considerations when it comes to the citizen’s social thought.

Which cereal do I feel like, says the consumer, and how much? How healthy are they? How good do they taste? How much do they cost? What is their fat, sugar, gluten, etc content? What about the brand? What about the important social information that I am never given, says the socially-aware consumer? What were the working conditions under which they were produced? Where were they made? How were they transported? What is their carbon footprint? Were animals harmed? Was there environmental damage?

This is an overload. It is an unnecessary overload. It’s too much data; usually any of the choices will be similar, or similarly bad. Sometimes there are a few excessively unhealthy or expensive or unethical choices, which can be quickly eliminated. Any of the others will be not much different from each other. It is ridiculous to devote more than a minute to deciding which breakfast cereal to choose, standing in the aisle. It may be less ridiculous to spend an hour researching which breakfast cereal is most socially beneficial — this information should be found in the supermarket, but is not. Nonetheless, the point remains that it would seem something quite a waste for an average citizen to be spending more than a miniscule fraction of their time deciding what to eat for breakfast.

With politics, it is different. Of course, the first thing to note is that comparing political action to consumer choices of breakfast cereal is odious and demeans the idea of citizenship. This of course has not stopped some social choice theorists from doing turning precisely that reduction into mathematics and calling it science.

But there are other differences between politics and breakfast cereal, other than their intrinsically different moral register. Even in the specifics, there are major differences. What do I think of these ideologies? What is my position on the State? On law? On political structures? Democracy? Private property? Organisation of work? Distribution of wealth? Economic mechanisms? This is a much, much greater overload — but it is a necessary one. Again, there are a few particularly unethical choices, which can be quickly eliminated — fascism, religious fundamentalism, colonialism, white supremacy, and so on.

Unlike other “paradoxes of choice”, it is not ridiculous to devote a serious amount of time to deciding what one thinks on these questions — and to use these answers to decide what to do politically. In fact it is necessary. The questions are complicated. The questions are important. The questions carry deep significance for our own individual and collective lives. And it’s not just a “choice” between existing alternatives. If it turns out that the “choice” (to the extent it exists) of what political ideas to follow, what actions to take, is a choice of the least bad alternative, then it is a bad set of choices. In this case, if every option is bad, one can alternatively try to do something *good* — one also has the option of imagining new ideas, building new alternatives, doing new things.

With the twenty brands of breakfast cereal, we can cut through the decision-making process by realising that, having thrown away obviously unacceptable answers, and in the absence of an obviously least bad choice, any of the remaining answers will probably be roughly equally acceptable. And in fact in practice one will tend to choose what was okay in the past.

With the hundred and more political issues to work on, one approach is similar, and perhaps one can sometimes cut through the question of what to do in a similar way. Sometimes there are obviously unacceptable answers — regressive, fundamentalist, sectarian, etc — and one can discard them. Sometimes there is an obvious choice given one’s own circumstances, talents and interests — for a computer hacker with a background in journalism, for an indigenous person living under Australian military “intervention”, for a member of an ethnic minority facing discrimination, there may be some choices that are clearly better than others. And if no such answer applies, after the clearly bad have been discarded, then probably any of the remaining choices will probably be positive, and beneficial to society. And in fact if we’ve been involved in one particular issue we may tend to continue with that, since we know more about it.

But this still seems like an inadequate way to think about it. Woefully inadequate.

* * *

Talk of “choice” is always problematic. Of course, for people in marginalised positions, suffering from various particular forms of oppression, if there is nothing to do but fight your oppressor, then you do it, there’s not much “choice” about it. To speak of “choice” in politics is to intrinsically speak about a privileged point of view. Nonetheless, in the richer world, with its wealth, cognitive surplus, and leisure time, there is significant choice of this type. And the choices are significant — they can help to determine what sort of world we live in, whether it is good or bad, whether it is worth living in, even whether it is alive or dead.

The inadequacy of the “paradox of choice” view of citizens thinking about their own social action, or activism, is that it conceptualises issues as separate “choices”. In fact what happens on one social issues affects several others, and they affect others, to that really all these “choices” are aspects of a single organic social whole. They are all connected.

And to be sure, at present at least, most of these “political choices” are choices to stem a regressive tide that pushes on all fronts. The choices are usually to curb one or another social evil — curb the tide of corporate power, reduce carbon emissions, stop the latest proposed war, pass a mild reform to ameliorate one of many possible outrages. A choice between rearguard actions, a choice between only defensive actions, is a choice between urgent and more urgent emergencies, a choice between living under outrageous or mildly less outrageous circumstances.

To be fair, many, or even most, defensive actions are inspired by hope for a better future, and do have the effect of building movements that can both fight negative change and then push for positive change.

But nonetheless, would it not be wonderful to have an actual positive choice, one that concretely builds the type of world you want to live in?

would it not be better to have an idea about the type of world you want to live in, and then measure each possible choice against it?

Would it not be better to have an idea about the type of world you want to live in, and then say — if each choice available is not particularly good, then try to build some aspect of that world you want to live in?

At the very least, would it not be at least helpful, as one thinks about what to do in the world, to have some ideas — however tenuous, however contingent, however conditional — about how the world could be, ought to be?

Having a reasonably well-formed — but not rigid, not sectarian, not a blueprint — idea of a good society is, on the one hand, a useful guide to deciding what to do now: will this action help move us towards a good, or better, society?

Having an idea about the type of world you want to live in can provide inspiration. And that applies regardless of whether one has the privilege of “choosing” what to do, or not.

On some fronts there may not be that far to go. To the extent you can already act in accordance with the world as it should be, the world is that much closer to utopia.

* * *

As to what positive political vision might consist of, that is in itself an important question. It must consist of more than vague slogans like “equal rights” or “freedom for all” or even slightly sharper notions like a “social economy” or “economic democracy”. I think it must say something about the shape of institutions in a better world, of what better institutions would look like, how they might operate.

It then entails criticism of proposed institutions, proposed visions — and no doubt this can descend into something that looks like political science fiction. Some of us like political science fiction, but not all of us do. It will take continual thought, continual rethinking and analysis, and experiment, to keep apart the line between, on the one hand, fact, proposed fact, and what is potentially factually possible, — and, on the other hand, fiction, or the impossible; though fiction of course has its merits, not least of stimulating the imagination. And of course it requires open-mindedness and flexibility.

But all such discussion, of course, is haunted by the spectre of communism. Not the spectre that haunted Europe in the writing of Marx and Engels, but the one that haunts the entire world in the wake of the horrors of the Soviet Union and other nations that called themselves “socialist”.

But history moves. That collapse was over 25 years ago now. It is time, historically, to pick up the pieces. The wounds have stopped bleeding. And even especially in the formerly “socialist” sixth of the earth is there a clear understanding of the horrors — different in kind, and less intense in many ways — of the current global economic system.

It has never been a logically good argument to respond to any thought of a radically better world to point to the Soviet Union. Yes, the first time humanity successfully overthrew capitalism, it was a disaster; but so what? History moves on, and there is no impossibility theorem that any non-capitalist system must be a disaster. On the contrary, capitalism is increasingly evidently a disaster, setting human society on a collision course with the biosphere’s physical limits. To many this has always been a disaster, and many have been making this argument, persuasively, for a long time.

At the very least, there is the converse: without any well-formed idea of what a good society would look like, there is a profound uncertainty about any political action whatsoever, a lack of inspiration, and uncertainty about what we stand for or why we should do anything. How do we know we are doing anything useful? What are we doing? Where are we going? Why should we do anything at all?

This seems to me to capture another portion of the apathy, the indifference, and the ennui in many societies today. And I think it goes to the heart of the problem.

While it may look like a paradox of choice, at its heart it is not. The real problem is not that we are overloaded with too many ideas about what to do. The real problem is that we do not have enough ideas about where we want to go.

Plane graphs, special alternating links, and contact geometry, Sydney Oct 2017

On Thursday October 5 2017 I gave a talk in the Geometry and Topology seminar at the University of Sydney.

The slides from the talk are available here.


Plane graphs, special alternating links, and contact geometry


There is a beautiful theory of polytopes associated to bipartite plane graphs, due to Alexander Postnikov, Tamas Kalman, and others. Via a construction known as the median construction, this theory extends to knots and links — more specifically, minimal genus Seifert surfaces for special alternating links. The complements of these Seifert surfaces also have interesting geometry. The relationships between these objects provide many interesting connections between graphs, spanning trees, polytopes, knot and link polynomials, and even Floer homology. In recent work with Kalman we showed how these connections extend to contact geometry. I’ll try to explain something of these ideas.


Tutte meets Homfly

Graphs are pretty important objects in mathematics, and in the world — what with every network of any kind being represented by one, from social connections, to road and rail systems, to chemical molecules, to abstract symmetries. They are a fundamental concept in understanding a lot about the world, from particle physics to sociology.

Knots are also pretty important. How can a loop of string get knotted in space? That’s a fairly basic question and you would think we might know the answer to it. But it turns out to be quite hard, and there is a whole field of mathematics devoted to understanding it. Lord Kelvin thought that atoms were knots in the ether. As it turns out, they are not, and there is no ether. Nonetheless the idea was an interesting one and it inspired mathematicians to investigate knots. Knot theory is now a major field of mathematics, part of the subject of topology. It turns out to be deeply connected to many other areas of science, because many things can be knotted: umbilical cords, polymers, quantum observables, DNA… and earphone cords. (Oh yes, earphone cords.) Indeed, knots arise crucially in some of the deepest questions about the nature of space and time, whether as Wilson loops in topological field theories, or as crucial examples in the theory of 3-dimensional spaces, or 3-manifolds, and as basic objects of quantum topology.

Being able to tell apart graphs and knots is, therefore, a pretty basic question. If you’re given two graphs, can you tell if they’re the same? Or if you’re given two knots, can you tell if they’re the same? This question is harder than it looks. For instance, the two graphs below look quite different, but they are the “same” (or, to use a technical term, isomorphic): each vertex in the first graph corresponds to a vertex in the second graph, in such a way that the connectedness of vertices is preserved.

(Source: Chris-martin on wikipedia)

Telling whether two graphs are the same, or isomorphic, is known as the graph isomorphism problem. It’s a hard problem; when you have very large graphs, it may take a long time to tell whether they are the same or not. As to precisely how hard it is, we don’t yet quite know.

Similarly, two knots, given as diagrams drawn on a page, it is difficult to tell if they are the “same” or not. Two knots are the “same”, or equivalent (or ambient isotopic), if there is a way to move one around in space to arrive at the other. For instance, all three knots shown below are equivalent: they are all, like the knot on the left, unknotted. This is pretty clear for the middle knot; for the knot on the right, have fun trying to see why!

(Source: Stannered, C_S, Prboks13 on wikipedia)

Telling whether two knots are equivalent is also very hard. Indeed, it’s hard enough to tell if a given knot is knotted or knot — which is known as the unknot recognition or unknotting problem. We also don’t quite know precisely how hard it is.

The fact that we don’t know the answers to some of the most basic questions about graphs and knots is part of the reason why graph theory and knot theory are very active areas of current research!

However, there are some extremely clever methods that can be used to tell graphs and knots apart. Many such methods exist. Some are easier to understand than others; some are easier to implement than others; some tell more knots apart than others. I’m going to tell you about two particular methods, one for graphs, and one for knots. Both methods involve polynomials. Both methods are able to tell a lot of graphs/knots apart, but not all of them.

The idea is that, given a graph, you can apply a certain procedure to write down a polynomial. Even if the same graph is presented to you in a different way, you will still obtain the same polynomial. So if you have two graphs, and they give you different polynomials, then they must be different graphs!

Similarly, given a knot, you can apply another procedure to write down a polynomial. Even if the knot is drawn in a very different way (like the very different unknots above), you still obtain the same polynomial. So if you have two knots, and they give you different polynomials, then they must be different knots!

Bill Tutte and his polynomial

Bill Tutte was an interesting character: a second world war British cryptanalyst and mathematician, who helped crack the Lorenz cipher used by the Nazi high command, he also made major contributions to graph theory, and developed the field of matroid theory.

He also introduced a way to obtain a polynomial from a graph, which now bears his name: the Tutte polynomial.

Each graph \(G \) has a Tutte polynomial \(T_G \). It’s a polynomial in two variables, which we will call \(x\) and \(y\). So we will write the Tutte polynomial of \(G\) as \(T_G (x,y)\).

For instance, the graph \(G\) below, which forms a triangle, has Tutte polynomial given by \(T_G (x,y) = x^2 + x + y\).

So how do you calculate the Tutte polynomial? There are a few different ways. Probably the easiest is to use a technique where we simplify the graph step by step in the process. We successively collapse or remove edges in various ways, and as we do so, we make some algebra.

There are two operations we perform to simplify the graph. Each of these two operations “removes” an edge, but does so in a different way. They are called deletion and contraction. You can choose any edge of a graph, and delete it, or contract it, and you’ll end up with a simpler graph.

First, deletion. To delete an edge, you simply rub it out. Everything else stays as it was. The vertices are unchanged: there are still just as many vertices. There is just one fewer edge. So, for instance, the result of deleting an edge of the triangle graph above is shown below.

The graph obtained by deleting edge \(e\) from graph \(G\) is denoted \(G-e\).

Note that in the triangle case above, the triangle graph is connected, and after deleting an edge, the result is still connected. This isn’t always the case: it is possible that you have a connected graph, but after moving an edge \(e\), it becomes disconnected. In this case the edge \(e\) is called a bridge. (You can then think of it as a “bridge” between two separate “islands” of the graph; removing the edge, there is no bridge between the islands, so they become disconnected.)

Second, contraction. To contract an edge, you imagine shrinking it down, so that it’s shorter and shorter, until it has no length at all. The edge has vertices at its endpoints, and so these two vertices come together, and combine into a single vertex. So if edge \(e\) has vertices \(v_1, v_2\) at its endpoints, then after contracting \(e\), the vertices \(v_1, v_2\) are combined into a single vertex \(v\). Thus, if we contract an edge of the triangle graph, we obtain a result something like the graph shown below.

The graph obtained by contracting edge \(e\) from graph \(G\) is denoted \(G.e\).

Contracting an edge always produces a graph with 1 fewer edges. Each edge which previous ended at \(v_1\) or \(v_2\) now ends at \(v\). And contracting an edge usually produces a graph with 1 fewer vertices: the vertices \(v_1, v_2\) are collapsed into \(v\).

However, this is not always the case. If the edge \(e\) had both its endpoints at the same vertex, then the number of vertices does not decrease at all! The endpoints \(v_1\) and \(v_2\) of \(e\) are then the same point, i.e. \(v_1 = v_2\), and so they are already collapsed into the same vertex! In this case, the edge \(e\) is called a loop. Contracting a loop is the same as just deleting a loop.

So, that’s deletion and contraction. We can use deletion and contraction to calculate the Tutte polynomial using the following rules:

  1. Let’s start with some really simple graphs.
    • If a graph \(G\) has no edges, then it’s just a collection of disconnected vertices. In this case the Tutte polynomial is given by \(T_G (x,y) = 1\).
    • If a graph has precisely one edge, then that it consists of a bunch of vertices, with precisely one edge \(e\) joining them. If \(e\) connects two distinct vertices, then it is a bridge, and \(T_G (x,y) = x\).
    • On the other hand, if \(G\) has precisely one edge \(e\) which connects a vertex to itself, then it is a loop, and \(T_G (x,y) = y\).
  2. When you take a graph \(G\) and consider deleting an edge \(e\) to obtain \(G – e\), or contracting it to obtain \(G.e\), these three graphs \(G, G-e, G.e\) have Tutte polynomials which are closely related:

    \(\displaystyle T_G (x,y) = T_{G-e} (x,y) + T_{G.e} (x,y)\).

    So in fact the Tutte polynomial you are looking for is just the sum of the Tutte polynomials of the two simpler graphs \(G-e\) and \(G.e\).
    However! This rule only works if \(e\) is not a bridge or a loop. If \(e\) is a bridge or a loop, then we have two more rules to cover that situation.

  3. When edge \(e\) is a bridge, then \(T_G (x,y) = x T_{G.e} (x,y)\).
  4. When edge \(e\) is a loop, then \(T_G (x,y) = y T_{G-e} (x,y)\).

These may just look like a random set of rules. And indeed they are: I haven’t tried to explain them, where they come from, or given any motivation for them. And I’m afraid I’m not going to. Tutte was very clever to come up with these rules!

Nonetheless, using the above rules, we can calculate the Tutte polynomial of a graph.

Let’s do some examples. We’ll work out a few, starting with simpler graph and we’ll work our way up to calculating the Tutte polynomial of the triangle, which we’ll denote \(G\).

First, consider the graph \(A\) consisting of a single edge between two vertices.

This graph contains precisely one edge, which is a bridge, so \(T_A (x,y) = x\).

Second, consider the graph \(B\) consisting of a single loop on a single vertex. This graph also contains precisely one edge, but it’s a loop, so \(T_B (x,y) = y\).

Third, consider the graph \(C\) which is just like the graph \(A\), consisting of a single edge between two vertices, but with another disconnected vertex.

This graph also contains precisely one edge, which is also a bridge, so it actually has the same Tutte polynomial as \(A\)! So we have \(T_C (x,y) = x\).

Fourth, consider the graph \(D\) which consists of a loop and another edge as shown. A graph like this is sometimes called a lollipop.

Now let \(e\) be the loop. As it’s a loop, rule 4 applies. If we remove \(e\), then we just obtain the single edge graph \(A\) from before. That is, \(D-e=A\). Applying rule 4, then, we obtain \(T_D (x,y) = y T_{D-e} (x,y) = y T_A (x,y) = xy\).

Fifth, consider the graph \(E\) which consists of two edges joining three vertices. We saw this before when we deleted an edge from the triangle.

Pick one of the edges and call it \(e\). (It doesn’t matter which one — can you see why?) If we remove \(e\), the graph becomes disconnected, so \(e\) is a bridge. Consequently rule 3, for bridges, applies. Now contracting the edge \(e\) we obtain the lollipop graph \(E\). That is, \(E-e=C\). So, applying rule 3, we obtain \(T_E (x,y) = x T_{E-e} (x,y) = x T_C (x,y) = x^2 \).

Sixth, let’s consider the graph \(F\) consisting of two “parallel” edges between two vertices. We saw this graph before when we contracted an edge of the triangle.

Pick one of the edges and call it \(e\). (Again, it doesn’t matter which one.) This edge is neither a bridge nor a loop, so rule 2 applies. Removing \(e\) just gives the graph \(A\) with one vertex, which has Tutte polynomial \(x\). Contracting \(e\) gives a graph with a single vertex and a loop. Applying rule 4, this graph has Tutte polynomial \(y\). So, by rule 2, the Tutte polynomial of this graph \(F\) is given by \(\displaystyle T_F (x,y) = x + y \).

Finally, consider the triangle graph \(G\). Take an edge \(e\); it’s neither a bridge nor a loop, so rule 2 applies. Removing \(e\) results in the graph \(E\) from above, which has Tutte polynomial \(x^2\). Contracting \(e\) results in the graph \(F\) from above with two parallel edges; and we’ve seen it has Tutte polynomial \(x+y\). So, putting it all together, we obtain the Tutte polynomial of the triangle as

\(\displaystyle T_G (x,y) = T_{G-e} (x,y) + T_{G.e} (x,y) = T_E (x,y) + T_F (x,y) = x^2 + x + y.\)

Having seen these examples, hopefully the process starts to make some sense.

However, as we mentioned before, we’ve given no motivation for why this works. And, it’s not even clear that it works at all! If you take a graph, you can delete and contract different edges in different orders and get all sorts of different polynomials along the way. It’s not at all clear that you’ll obtain the same result regardless of how you remove the edges.

Nonetheless, it is true, and was proved by Tutte, that no matter how you simplify the graph at each stage, you’ll obtain the same result. In other word, the Tutte polynomial of a graph is actually well defined.


H, O, M, F, L, Y, P and T

Tutte invented his polynomial in the 1940s — it was part of his PhD thesis. So the Tutte polynomial has been around for a long time. The knot polynomial that we’re going to consider, however, is considerably younger.

In the 1980s, there was a revolution in knot theory. The excellent mathematician Vaughan Jones in 1984 discovered a polynomial which can be associated to a knot. It has become known as the Jones polynomial. It was not the first polynomial that anyone had defined from a knot, but it sparked a great deal of interest in knots, and led to the resolution of many previously unknown questions in knot theory.

Once certain ideas are in the air, other ideas follow. Several mathematicians started trying to find improved versions of the Jones polynomial, and at least 8 mathematicians came up with similar ways to improve the Jones polynomial. In 1985, Jim Hoste, Adrian Ocneanu, Kenneth Millett, Peter J. Freyd, W. B. R. Lickorish, and David N. Yetter published a paper defining a new polynomial invariant. Making an acronym of their initials, it’s often called the HOMFLY polynomial. Two more mathematicians, Józef H. Przytycki and Pawe? Traczyk, did independent work on the subject, and so it’s often called the HOMFLY-PT polynomial.

Like the Tutte polynomial, the HOMFLY polynomial is a polynomial in two variables. (The Jones polynomial, however, is just in one variable.) It can also be written as a homogeneous polynomial in three variables. We’ll take the 3-variable homogeneous version.

Strictly speaking, to get a HOMFLY polynomial, your knot must be oriented: it must have a direction. This is usually represented by an arrow along the knot.

HOMFLY polynomials also exist for links — a link is just a knot with many loops invovled. So even if there are several loops knotted up, they still have a HOMFLY polynomial. (Each loop needs to be oriented though.)

So, if you’re given an oriented knot or link \(K\), it has a HOMFLY polynomial. We’ll denote it by \(P_K (x,y,z)\). So how do you compute it? By following some rules which successively simplify the knot.

  1. If the knot \(K\) is the unknot, then \(P_K (x,y,z) = 1\).
  2. If you take one of the crossings in the diagram and alter it in the various ways shown below — but leave the rest of the knot unchanged — then you obtain three links \(L^+, L^-, L^0\). Their HOMFLY polynomials are related by

    \(\displaystyle x P_{L^+} (x,y,z) + y P_{L^-} (x,y,z) + z P_{L^0} (x,y,z) = 0\).

    Source: C_S, wikimedia

    A relationship like this, between three knots or links which differ only at a single crossing, is called a skein relation.

  3. If you can move the link \(L\) around in 3-dimensional space to another link \(L’\), then this doesn’t change the HOMFLY polynomial: $latex P_L (x,y,z) = P_{L’} (x,y,z).
  4. If the oriented link \(L\) is split, i.e. separates into two disjoint (untangled) sub-links \(L_1, L_2\), then you can take the HOMFLY polynomials of the two pieces separately, and multiply them, with an extra factor:

    \(\displaystyle P_L (x,y,z) = \frac{-(x+y)}{z} P_{L_1} (x,y,z) P_{L_2} (x,y,z)\).

  5. If \(L\) is a connect sum of two links \(L_1, L_2\), then you can take the HOMFLY polynomials of the two pieces, and multiply them:

    \(\displaystyle P_L (x,y,z) = P_{L_1} (x,y,z) P_{L_2} (x,y,z)\).

    What is a connect sum? It’s when two knots or links are joined together by a pair of strands, as shown below.

    Source: Maksim, wikimedia

And there you go.

Again, it’s not at all clear where these rules come from, or that they will always give the same result. There might be many ways to change the crossings and simplify the knot, but H and O and M and F and L and Y and P and T showed that in fact you do always obtain the same result for the polynomial.

Let’s see how to do this in a couple of examples.

First of all, for the unknot \(U\), by rule 1, its HOMFLY polynomial is \(P_U (x,y,z) = 1\).

Second, let’s consider two linked unknots as shown below. This is known as the Hopf link. Let’s call it \(H\).

Source: Jim.belk, wikimedia.

Let’s orient both the loops so that they are anticlockwise. Pick one of the crossings and consider the three possibilities obtained by replacing it according to the skein relation described above, \(H^+, H^-, H^0\). You should find that \(H^+\) corresponds to the crossing as it is shown, so \(H^+ = H\). Changing the crossing results in two unlinked rings, that is, \(H^- =\) two split unknots. By rule 4 above then, \(P_{H^-} (x,y,z) = \frac{-(x+y)}{z} P_U (x,y,z) P_U (x,y,z)\); and as each unknot has HOMFLY polynomial \(1\), we obtain \(P_{H^-} (x,y,z) = \frac{-(x+y)}{z}\). On the other hand, smoothing the crossing into \(H^0\) gives an unknot, so \(P_{H^0} (x,y,z) = P_{U} (x,y,z) = 1\).

Putting this together with the skein relation (rule 2), we obtain the equation

\(\displaystyle x P_{H^+} (x,y,z) + y P_{H^-} (x,y,z) + z P_{H^0} (x,y,z) = 0\),

which gives

\(\displaystyle x P_H (x,y,z) + y \frac{-(x+y)}{z} + z = 0\)

and hence the HOMFLY of the Hopf link is found to be

\(\displaystyle P_H (x,y,z) = \frac{ y(x+y)}{xz} – \frac{z^2}{xz} = \frac{xy + y^2 – z^2}{xz}\).


When the Tutte is the HOMFLY

In 1988, Francois Jaeger showed that the Tutte and HOMFLY polynomials are closely related.

Given a graph \(G\) drawn in the plane, it has a Tutte polynomial \(T_G (x,y)\), as we’ve seen.

But from such a \(G\), Jaeger considered a way to build an oriented link \(D(G)\). And moreover, he showed that the HOMFLY polynomial of \(D(G)\) is closely related to the Tutte polynomial of \(G\). In other words, \(T_G (x,y)\) and \(P_{D(G)} (x,y,z)\) are closely related.

But first, let’s see how to build an link from a graph. It’s called the median construction. Here’s what you do. Starting from your graph \(G\), which is drawn in the plane, you do the following.

  • Thicken \(G\) up. You can then think of it as a disc around each vertex, together with a band along each edge.
  • Along each edge of \(G\), there is now a band. Take each band, and put a full right-handed twist in it. You’ve now got a surface which is twisted up in 3-dimensional space.
  • Take the boundary of this surface. It’s a link. And this link is precisely \(D(G)\). (As it turns out, there’s also a natural way to put a direction on \(D(G)\), i.e. make it an oriented link.)

It’s easier to understand with a picture. Below we have a graph \(G\), and the link \(D(G)\) obtained from it.

A graph (source: wikimedia) and the link (source: Jaeger) obtained via the median construction.

Jaeger was able to show that in general, the Tutte polynomial \(T_G (x,y)\) and the HOMFLY polynomial \(P_{D(G)} (x,y,z)\) are related by the equation

\(\displaystyle P_{D(G)} (x,y,z) = \left( \frac{y}{z} \right)^{V(G)-1} \left( – \frac{z}{x} \right)^{E(G)} T_G \left( -\frac{x}{y}, \frac{-(xy+y^2)}{z^2} \right),\)

where \(V(G)\) denotes the number of vertices of \(G\), and \(E(G)\) denotes the number of edges of \(G\). Essentially, Jaeger showed that the process you can use to simplify the link \(D(G)\) to calculate the HOMFLY polynomial, corresponds in a precise way to the process you can use to simplify the graph \(G\) to calculate the Tutte polynomial.

In addition to this excellent correspondence — Tutte meeting HOMFLY — Jaeger was able to deduce some further consequences.

He showed that the four colour theorem is equivalent to a fact about HOMFLY polynomials: for every loopless connected plane graph \(G\), \(P_{D(G)} (3,1,2) \neq 0\).

Moreover, since colouring problems for plane graphs are known to be very hard, in terms of computational complexity — NP-hard — it follows that the computation of the HOMFLY polynomial is also NP hard.

Said another way: if you could find a way to compute the HOMFLY polynomial of a link in polynomial time, you would prove that \(P = NP\) and claim yourself a Millennium prize!

Polytopes, dualities, and Floer homology

(41 pages)

on the arXiv.

Abstract: This article is an exposition of a body of existing results, together with an announcement of recent results. We discuss a theory of polytopes associated to bipartite graphs and trinities, developed by Kálmán, Postnikov and others. This theory exhibits a variety of interesting duality and triality relations, and extends into knot theory, 3-manifold topology and Floer homology. In recent joint work with Kálmán, we extend this story into contact topology and contact invariants in sutured Floer homology.

Local pdf available here (703 kb).


Adani: icon of Australian climate infamy

Here we are, in the year 2017. With now 25 years of climate-change international agreements behind us, here we are still trying to build oil pipelines and coal mines.

It is sad. Sad for humanity.

It is no longer a question of reducing the speed at which we are approaching the cliff. It is now a question of counting the metres to the cliff, as it approaches so fast.  It is no longer a question of reducing climate emissions to a reasonable level. It is now a question of counting the remaining tons which may be emitted, budgeting them carefully and switching off them with an emergency.

There are various different ways the budget can be calculated. The MCC Carbon Clock calculates that to limit increase in global average temperature to 2 degrees Celsius, the CO2 budget remaining is 734 gigatonnes, on a moderate (neither optimistic nor pessimistic) set of assumptions. At the current rate of emissions, that budget will be exhausted by 2035. That is time at which, continuing as we are now, scientific laws predict failure.

But there is a strong argument that a 2 degrees Celsius increase is too much. It means a vast range of climate impacts – for instance, at 2 degrees tropical coral reefs do not stand a chance. A limit of 1.5 degrees increase is an altogether better goal. Indeed, the Paris agreement aims to hold increase in global average temperature to 2 degrees Celsius, but “to pursue efforts” to limit the increase to 1.5 degrees Celsius. And the MCC Carbon Clock, under the same set of assumptions, calculates the remaining CO2 budget, to limit the increase to 1.5 degrees Celsius, as 41.3 gigatonnes. At the current rate, this budget will be exhausted in September 2018 — in just over a year’s time.

One year. Just one year.

Either way, it is a clear and present danger, urgent, all-absorbing, putting all other tumults to silence. All efforts must be to get off fossil fuels immediately, now, yesterday.

Yet where are we in Australia? We are about to build our biggest coal mine ever.

Adani, corrupt, lawbreaking Adani. They still want to build their Carmichael mine in the Galilee Basin in Queensland.

And Australian governments still fall over themselves to assist them. Approvals and re-approvals flow from the federal Liberal government. The Queensland Labor premier’s interventions have convinced Adani to go ahead. The total emissions of the Carmichael project — producing and burning the coal – will be 4.7 gigatonnes of CO2. That’s over 10% of the remaining budget for 1.5 degree increase. Just this one mine.

The project is itself barely financially viable. Adani says a special loan from the Northern Australian Infrastructure Facility (NAIF) is critical to their financing. Nonetheless contracts were announced in July. They say it will employ 10,000 people: the reality, as given by Adani’s own expert under oath, is closer to 1,500.

Like everything else in Australia, the mine would be built on Aboriginal land. The traditional owners, the Wangan and Jagalingou people, released a statement: Stop Adani destroying our land and culture.

If the Carmichael mine were to proceed it would tear the heart out of the land. The scale of this mine means it would have devastating impacts on our native title, ancestral lands and waters, our totemic plants and animals, and our environmental and cultural heritage. It would pollute and drain billions of litres of groundwater, and obliterate important springs systems. It would potentially wipe out threatened and endangered species. It would literally leave a huge black hole, monumental in proportions, where there were once our homelands. These effects are irreversible. Our land will be “disappeared”.

Native Title claims being too much of an uncertain quantity for Adani – and courts showing an increasing level of respect for indigenous desires to control their land — legislation was dutifully passed by the federal parliament in June to smooth Adani’s way.

Just like the Keystone XL and Dakota Access pipelines in the US, which have seen such inspiring resistance, if the Adani Carmichael mine is built it will be game over for the climate.

It cannot go ahead. It must not.

Adani continues to acquire property along its proposed rail corridor, even as new accusations of fraud emerge against them. But largely they are currently playing a waiting game. The minister responsible for NAIF, Matt Canavan, stepped down over the citizenship farce; and his replacement is Barnaby Joyce. Until the High Court rules, and possibly byelections are held, it may wait.

That gives a crucial opportunity to press the opposition to the mine and decisively stop it. Protests continue.  A few days ago, religious leaders promised civil disobedience.

As these leaders argued, it is a simple moral choice.

It is a simple scientific choice too.

Stop Adani’s mine, and switch this sunburnt country to renewable energy now.